Market Structure and Competition
Besanko and Braeutigam, CH 13
Western University
Chapter 13 Overview
- What happens when there is more than one seller, but still there are few enough to affect the market?
- How do their choices affect each other?
- Will they cooperate? Will they compete?
- What happens if their products are imperfect substitutes?
Objectives
- Describing and Measuring Market Structure
- Oligopoly with Homogeneous Products
- Dominant Firm Markets
- Oligopoly with Horizontally Differentiated Products
- Monopolistic Competition
- The Cournot Equilibrium and the Inverse Elasticity Pricing Rule
Market Structures: 2 Key Dimensions
| Market Structure | Number of Firms | Type of Product | Control Over Price | Examples |
|---|---|---|---|---|
| Perfect Competition | Many | Homogeneous | None | Agriculture (US) |
| Monopolistic Competition | Many | Differentiated | Some | Retail stores |
| Oligopoly | Few | Homogeneous or Differentiated | Some to Significant | Banking-Big 5 (CA) |
| Monopoly | One | Unique | Significant | Utilities |
| Dominant Firm | One dominant, Many small | Homogeneous | Significant by dominant firm | US: Scotch Tape (3M) |
Measures of Market Structure
Four-Firm Concentration Ratio (4CR): The share of industry sales revenue accounted for by the four firms with the largest sales revenue in the industry
Herfindahl–Hirschman Index (HHI): The sum of the squares of the market share of each firm in the industry (\(0 \le HHI \le 10,000\))
Market Structure 4CR HHI Perfect Competition Low Low Olygopoly Intermediate Intermediate Monopoly 100 10,000
Oligopoly with Homogeneous Goods
- A central feature of oligopoly markets: competitive interdependence
- The decisions of every firm significantly affect the profits of competitors
- Perfect Competition: No impact of one firm on its rivals
- Monopolist: No rivals
- Central question of oligopoly theory: how does the close interdependence among firms in the market affect their behavior?
The Cournot Model
- 2 firms (duopoly); one homogeneous good
- Both with identical marginal costs
- Firms choose output (how much to produce)
- simultaneously
- non-cooperatively (no collusion)
- no knowledge of each other’s plan
The Cournot Model
- Inverse demand is downward-slopping and a function of the combined output of the two firms, \(P(Q_1+Q_2)\)
- Price is not known until both firms have made their output choice
- Each firm will produce the output choice that maximizes its profit based on its expectation of the other firm’s output choice
Residual Demand
- Suppose firm 1 expects that firm 2 will produce \(Q^e_2\)
- If firm 1 produces \(Q_1\), then total output will be \(Q=Q_1+Q_2^e\), and
- and market price will be \(P(Q)=P(Q_1+Q_2^e)\)
- the inverse market demand resulting from holding their rivals’ output constant is the residual demand
- Ex. Linear Demand \[ P(Q)=a-b(Q_1+Q_2^e)=(a-bQ_2^e)-bQ_1 \]
Residual Demand
Example
- Inverse demand curve \(P(Q)=100-Q\)
- \(MC=10\)
- Firm 1 is Samsung and Firm 2 is SK
- If \(Q=80\), then \(P(80)=20\)
- If \(Q^e_2=50\), then \(P(Q_1+50)=(100-50)-Q_1\)
- for \(Q_1=30\), \(P(30+50)=20\)
Residual Demand
Best Response
When choosing output, each firm will act as a monopolist relative to its residual demand
Firm 1 maximizes its profit by choosing \(Q_1\) that maximizes: \[ \pi_1=P(Q_1+Q_2^e)⋅Q_1−C(Q_1) \]
Best Response: For any given belief about the output of firm 2, \(Q^e_2\), there is an optimal choice of output for firm 1, \[ Q_1=BR_1(Q^e_2) \]
Reaction Function
The best response function for Firm 1, \(BR_1(Q_2^e)\), is derived by setting: \[ \frac{\partial \pi_1}{\partial Q_1} = 0 \]
Example with Linear Demand: Assuming \(P(Q) = a - b(Q_1 + Q_2)\) and constant marginal cost \(MC = c\), the reaction functions can be simplified as: \[ \begin{aligned} Q_1= \frac{a - c}{2b} - \frac{Q_2^e}{2} \;;\; Q_2= \frac{a - c}{2b} - \frac{Q_1^e}{2} \end{aligned} \]
Reaction Function
Example (continued)
- With \(P(Q)=100-Q\) and \(MC=10\)
- \[ Q_1=45-\frac{Q^e_2}{2} \;;\;Q_2=45-\frac{Q^e_1}{2} \]
Rection Function
- \(Q_1(Q^e_2)=45-\frac{Q^e_2}{2}\)
- \(Q_1(50)=20\)
- \(Q_1(30)=30\)
- \(Q_1(20)=35\)
Cournot Equilibrium
- Cournot Equilibrium: each firm maximizes its profits, given its beliefs about the other firm’s output choice, and those beliefs are confirmed in equilibrium
- Each firm optimally produces the output its rival expects it to produce
- No firm has incentives to deviate
- Mathematically, a combination of output choices \((Q_1^*,Q_2^*)\) that satisfy \[ \begin{aligned} Q^*_1= \frac{a - c}{2b} - \frac{Q_2^*}{2} \;;\; Q^*_2= \frac{a - c}{2b} - \frac{Q_1^*}{2} \end{aligned} \]
Cournot Equilibrium
Cournot Equilibrium for Two Firms: Occurs where the reaction curves of Firm 1 and Firm 2 intersect.
Equilibrium Output: Solving the system of equations given by the reaction functions yields the equilibrium outputs for both firms: \[ Q_1^* = Q_2^* = \frac{a - c}{3b} \] and the total market output is: \[ Q^* = Q_1^* + Q_2^* = \frac{2(a - c)}{3b} \]
Example (continued)
Continuing with our example \(P=100-Q\) and \(MC=10\)
Firm output is \(Q_i=30\), total market output is \(Q=60\)
So, market price is \(P=40\)
Achieving Equilibrium
In the Cournot equilibrium both firms fully understand their interdependence and have confidence in each other’s rationality
Cournot for N-firms
For \(N\) identical firms, total output is \(Q=\sum_i^N Q_i\)
Firm \(i\) solves \[ \max_{Q_i} P(Q)⋅Q_i-C(Q_i) \qquad(1)\]
For linear demand, and \(MC=c\), each firm \(i\) will produce \[ Q^*_i = \frac{1}{N+1}\frac{a - c}{b} \]
Cournot for N-firms
- Total market output and market price will be \[ Q^*=\sum_i Q^*_i = \frac{N}{N+1}\frac{(a - c)}{b} \] \[ P^*=a-b\left(\frac{N}{N+1}\frac{(a - c)}{b} \right) = \frac{a+Nc}{(N+1)} \]
Comparing Market Structures
| Market Structure | \(N\) | Optimal Firm Output \(Q^*_i\) | Total Market Output \(Q^*\) | Market Price \(P^*\) |
|---|---|---|---|---|
| Monopoly | 1 | \(\frac{a - c}{2b}\) | \(\frac{a - c}{2b}\) | \(\frac{a + c}{2}\) |
| Cournot Duopoly | 2 | \(\frac{a - c}{3b}\) | \(\frac{2(a - c)}{3b}\) | \(\frac{(a + 2c)}{3}\) |
| Perfect Competition | \(\infty\) | 0 (virtually) | \(\frac{a - c}{b}\) | \(c\) |
Comparing Market Structures
- Monopoly: A single firm controls the entire market, producing half the competitive output and selling at a higher price.
- Cournot Duopoly: Two firms share the market, leading to increased output and lower prices compared to monopoly.
- Perfect Competition: An infinite number of firms produce at marginal cost, leading to the highest output and lowest price.
Comparing Market Structures
Cournot vs Monopoly
By independently maximizing their own profits, firms produce more total output than they would if they collusively maximized industry profits (Monopoly).
Cournot and Elasticity of Demand
FOC of Equation 1 gives \[ P(Q)+\frac{\partial P}{\partial Q}⋅Q_i=MC(Q_i) \]
It can be shown that \[ P(Q)\left[1-\frac{1}{|\epsilon_{Q,P}|/s_i}\right]=MC(Q_i) \] where \(s_i=Q_i/Q\) is the market share
Cournot and Elasticity of Demand
- \(|\epsilon_{Q,P}|/s_i\), the elasticity of the demand curve facing the firm:
- the smaller the market share of the firm, the more elastic the demand curve it faces
\(s_i\) Demand Curve Condition Market Structure 0 Flat \(P=MC\) Perfect Competition 1 Market Demand \(P(1-\frac{1}{|\epsilon_{Q,P}|})=MC\) Monopoly
Bertrand Competition
Simultaneous Price Setting
- Price Competition: Unlike Cournot, where firms compete by choosing quantities, in the Bertrand model, firms compete by setting prices and letting the market determine the quantity sold. Simultaneous, non-cooperative decisions.
- Homogeneous Products: Firms produce identical products, making price the sole factor for consumers when choosing between firms
- Cost Structures: Firms have identical cost structures and sufficient capacity to meet all demand
- Rational Expectations: Firms are profit-maximizers and have rational expectations about their competitors’ behavior
Bertrand Model
- Profit Maximization for Firm 1: \[
\pi_1 = P_1 \cdot Q_1(P_1, P_2) -C(Q_1(P_1,P_2))
\]
- where \(P_1\) is the price set by Firm 1,
- \(C(Q_1(P_1,P_2))\) is cost function for Firm 1, and
- \(Q_1(P_1, P_2)\) is the demand for Firm 1’s product as a function of both firms’ prices
Demand Scenarios for Firm 1
- When \(P_1 = P_2\):
- If both firms set equal prices, we assume demand is split equally between them. Hence, \(Q_1 =Q_2 = \frac{1}{2}Q(P_1)\), where \(Q(P_1)\) is the total market demand at price \(P_1\)
- When \(P_1 > P_2\):
- Firm 1 captures none of the market demand since consumers prefer the cheaper, identical product from Firm 2. Thus, \(Q_1 = 0\)
- When \(P_1 < P_2\):
- Firm 1 captures the entire market demand because its price is lower. Hence, \(Q_1 = Q(P_1)\)
Residual Demand Bertrand
We can summarize the previous analysis as \[ Q_1(P_1,P_2) = \begin{cases} 0 ,& P_1>P_2 \\ \frac{1}{2}Q(P_1),& P_1=P_2 \\ Q(P_1) ,& P_1 < P_2 \end{cases} \]
Residual Demand Bertrand
Example (continued)
Bertrand Equilibrium
Definition: The Bertrand equilibrium is reached when each firm sets its price such that it maximizes its profits and neither firm can increase profits by unilaterally changing its price.
Reaching the Bertrand Equilibrium:
- With identical products and rational, profit-maximizing firms, the equilibrium occurs when both firms set their prices equal to marginal cost (\(P = MC\)).
- If one firm undercuts the other by even a small amount, it captures the entire market. Thus, to avoid losing the market, firms race to the bottom, stopping at \(P = MC\).
Bertrand Equilibrium
- Characteristics:
- The Bertrand equilibrium results in the competitive price level, even with only two firms in the market.
- Total market supply meets all demand at this price level, as firms have sufficient capacity.
Bertrand vs. Cournot
- In Cournot competition, firms choose quantities,
- leading to higher prices than marginal costs, \(P>MC\), due to the strategic reduction in total output.
- In Bertrand competition, firms choose prices,
- leading to a price equal to marginal cost, \(P=MC\), similar to perfect competition outcomes, even with only two firms.
Bertrand and Perfect Competition
- Both result in prices equal to marginal costs, \(P=MC\);
- however, perfect competition assumes an infinite number of firms, \(N=\infty\),
- while Bertrand demonstrates that two firms \(N\ge2\), are sufficient to reach competitive pricing under price competition.
Stackelberg Model
Quantity Leadership
- Sequential Moves: Unlike Cournot and Bertrand models, the Stackelberg model assumes that firms move sequentially. One firm, designated as the leader, chooses its output first; the follower firm then reacts to this decision.
- Homogeneous Products: Firms produce identical products.
- Known Reaction Function: The leader firm is aware of the follower firm’s reaction function and uses this information to optimize its own output decision.
- Complete Information: Both firms have complete information about the market, including costs and demand.
The Follower’s Problem
- General Framework: The follower firm chooses its output \(Q_2\) to maximize its profit, given the output \(Q_1\) of the leader firm. The follower’s profit function is: \[ \pi_2(Q_2) = P(Q_1 + Q_2)Q_2 - C_2(Q_2) \] where \(P(Q)\) is the market price function and \(C_2(Q_2)\) represents the cost of producing \(Q_2\).
Deriving the Reaction Function
- The profit-maximization choice of the follower firm will depend on the leader’s choice \(Q_1\)
- The reaction function \(Q_2 = BR_2(Q_1)\) shows this relationship
- \(BR_2(Q_1)\) derived by solving: \[ \frac{d\pi_2}{dQ_2} = 0 \] This equation gives the best response of the follower to any given \(Q_1\) by the leader.
Follower’s Reaction Curve
Assumptions: Linear inverse demand curve \(P(Q) = a - b(Q_1 + Q_2)\) and constant marginal costs \(MC_1 = MC_2 = c\).
Follower’s Reaction Function: With these assumptions, the follower’s reaction function simplifies to: \[ Q_2 = \frac{a - c - bQ_1}{2b} \qquad(2)\] This function indicates how the follower’s optimal output varies with the leader’s output level.
Isoprofit Curves
Isoprofit curves represent combinations of \(Q_1\) and \(Q_2\) that yield the same profit level for the follower. These curves help illustrate the follower’s best response to different leader outputs.
In the case of linear demand and constant marginal costs, isoprofit curves are comprised of all points (\(Q_1,Q_2\)) that satisfy equations of the form \[ \bar\pi_2=aQ_2-bQ_1Q_2-bQ_2^2 \]
Reaction and Isoprofit Curves
Reaction and Isoprofit Curves
- Profits of firm 2 will increase as we move to isoprofit lines that are closer to the left, (\(Q_1\) decreases)
- Highest profits when firm 2 is a monopolist (\(Q_1=0\))
- Profit-maximization implies that for each choice of \(Q_1\), firm 2 will pick the value of \(Q_2\) that reaches the isoprofit curve furthest to the left
- It follows that the slope of the isoprofit line must be vertical at the optimal choice (tangency condition)
- The locus of these tangencies form \(BR_2(Q_1)\)
The Leader’s Problem
Leader’s Objective: Given the follower’s reaction function, the leader firm optimizes: \[ \begin{aligned} \max_{Q_1}\pi_1(Q_1) &= P(Q_1 + Q_2)Q_1 - C_1(Q_1) \\ &\text{s.t. } Q_2=BR_2(Q_1) \end{aligned} \]
The leader internalizes that its actions influence the output of the follower
- total output \(Q=Q_1+BR_2(Q_1)\)
The Leader’s Problem
- With linear demand and constant marginal costs \[ \begin{aligned} \max_{Q_1}\pi_1(Q_1) &= (a - b(Q_1 + Q_2))Q_1 - cQ_1 \\ &\text{s.t. } Q_2=\frac{a - c - bQ_1}{2b} \end{aligned} \]
The Leader’s Problem
- Substituting the follower’s reaction function into the leader’s profit function and simplifying we get \[ \max_{Q_1}\pi(Q_1)=\frac{a-c}{2}Q_1-\frac{b}{2}Q_1^2 \]
Optimal Quantities
- Solving the leader’s maximization problem yields the Stackelberg equilibrium quantities for both firms:
- For the leader \(Q_1^*=\frac{a-c}{2b}\)
- and the follower, substituting into the \(BR_2(Q_1),\) \(Q_2^*=\frac{a-c}{4b}\)
- Total output \(Q^*=\frac{3(a-c)}{4b}\)
- Market price \(P^*(Q)=\frac{a+3c}{4}\)
Stackelberg Equilibrium
Stackelberg Equilibrium
- The follower will choose an output along its reaction curve \(BR_2(Q_1)\)
- The leader chooses an output combination that yields the highest profits
- Thus, the leader chooses output such that the isoprofit curve is furthest down and tangent to the firm’s 2 best reaction curve
- In other words, firm 1 selects the point along \(BR_2(Q_1)\) that reaches the highest isoprofit curve;
- The reaction curve is tangent to the isoprofit curve at this point
Discussion
- The leader’s advantage of moving first allows it to secure a more favorable portion of the market, influencing the overall market structure and outcomes
- Unlike the Cournot outcome, which was symmetric, under the Stackelberg outcome, the leader produces more output than the follower (exactly twice as much in fact).
- Even though the market price is lower under the Stackelberg outcome than under the Cournot outcome, the leader’s profit under the Stackelberg outcome is higher than its profit at the Cournot equilibrium.
- An oligopolist benefits by choosing its output first (sequential game)
| Market Structure | \(N\) | Optimal Firm Output \(Q^*_i\) | Total Market Output \(Q^*\) | Market Price \(P^*\) |
|---|---|---|---|---|
| Monopoly | 1 | \(\frac{a - c}{2b}\) | \(\frac{a - c}{2b}\) | \(\frac{a + c}{2}\) |
| Cournot Duopoly | 2 | \(\frac{a - c}{3b}\) | \(\frac{2(a - c)}{3b}\) | \(\frac{(a + 2c)}{3}\) |
| Stackelberg | 2 | Leader \(\frac{a-c}{2b}\) ; Follower \(\frac{a-c}{4b}\) | \(\frac{3(a-c)}{4b}\) | \(\frac{a+3c}{4}\) |
| Perfect Competition | \(\infty\) | 0 (virtually) | \(\frac{a - c}{b}\) | \(c\) |
Dominant Firm Market
Price Leadership Model
- Dominant Firm: One firm has a large enough market share to set the price. This firm is the price leader.
- Follower Firm:
- In equilibrium, the follower must always set the same price as the leader (follows from homogeneous product assumption)
- Followers take the price set by the dominant firm and choose their profit-maximizing output (competitive behavior)
- Homogeneous Products: The products offered by the dominant firm and the competitive fringe are identical, making price the primary competitive tool.
- Market Demand and Supply: The dominant firm considers the residual demand curve, which is the market demand curve minus the supply of the competitive fringe.
The Follower’s Problem
Objective: The follower firm aims to maximize its profits given the price set by the leader. The follower assumes this price as fixed and decides its quantity accordingly.
Profit Maximization: \[ PQ_2 - C_2(Q_2) \] where \(P\) is the price set by the leader, and \(C_2(Q_2)\) is the cost function for the follower.
Output Decision: The follower determines its output level \(Q_2\) where price equals marginal cost (\(P = MC_2\)), leading to a supply curve for the follower \(S(P)\).
The Leader’s Problem in Price Leadership
Forecasting Follower’s Response: The leader sets a price \(P\) anticipating the follower’s response. The leader must solve for the price that maximizes its profit, considering the residual demand it faces after the follower’s supply.
Residual Demand Curve: The leader’s demand is the market demand minus the follower’s supply, \(R(P) = D(P) - S(P)\).
Profit Maximization for the Leader: The leader’s profit is \(\pi_1(P) = (P - c)[D(P) - S(P)] = (P - c)R(P)\). The leader aims to choose \(P\) where marginal revenue from the residual demand curve equals marginal cost.
Example
- Demand is \(D(P) = \frac{a}{b}- \frac{P}{b}\),
- the follower’s cost function is quadratic, \(C_2(Q_2)=Q^2_2/2\) and
- the leader’s cost function is \(C_1(Q_1)=cQ_1\)
- Solving the follower’s problem we can derive the follower’s supply curve \[ P=Q_2 \implies S(p)=P \]
Example
- Residual demand for the leader \[ R($P)=D(P)-S(P)=\frac{a}{b}- \frac{P}{b}-P=\frac{a}{b}- \frac{b+1}{b}P \]
- Solving for P, \(P=\frac{a}{b+1}-\frac{b}{b+1}Q_1\)
- Leader’s Optimal Price and Output: By setting marginal revenue equal to marginal cost for the residual demand curve, \(Q_1^*=\frac{a-c(b+1)}{2b}\); \(P^*=\frac{a-c(b+1)}{2(b+1)}\)
Comparison with Other Market Models
- Distinctive Features of Price Leadership:
- The dominant firm has the ability to influence market prices directly, unlike in Cournot or Bertrand models where firms’ strategies are interdependent without a clear leader.
- Unlike the Stackelberg model, where leadership is defined by the sequence of moves, price leadership establishes dominance through market share and the ability to set prices.
- Market Efficiency and Welfare:
- The price leadership model can result in higher prices and lower outputs compared to perfectly competitive markets but may be more efficient than monopolistic outcomes due to the presence of the competitive fringe.